Our results confirm in the Riemannian setting the classical planar results
of Rubistein & Sternberg [1]}, Jerrard & Soner
[2]}, F.H. Lin
[3]}, [4]} and Sandier & Serfaty
[5]}.
Quite recently Chen & Sternberg [6]} have proved the convergence of the (rescaled) heat flow of the Ginzburg-Landau energies
defined on maps \(u:M\rightarrow \mathbb {C}\)  where \(M\)  is a two dimensional compact simply connected Riemannian manifold without boundary.
Our setting and our results are however different for some crucial points.
As we deal with vector fields, the topology of \(M\)  triggers the formation of the vortices via the Poincaré-Hopf Theorem, while in [6]} the vortices arise thanks to the choice of initial conditions and the scaling of the equation.
Another difference lies in the energy or, more precisely, in the Dirichlet part of the energy.
The energy \(\frac{1}{2}\int _{M}\left| \nabla u \right|^2_g \mathrm {vol}_g\)  defined on maps \(u:M\rightarrow \mathbb {C}\)  (in this case \(\nabla \)  stands for the Riemannian gradient) is conformally invariant. This is not the case for the energy \(\frac{1}{2}\int _{M}\left| D u \right|^2_g \mathrm {vol}_g\)  defined on vector fields (as before \(D\)  is the covariant derivative).
The conformal invariance combined with the Uniformization Theorem permits (as in the “static” situation considered in [8]}) to transfer the analysis from the manifold \(M\)  to a reference domain in \(\mathbb {R}^2\) .
As a result, the renormalized energy is modeled on the Laplacian in \(\mathbb {R}^2\)  with corrections due to the conformal change. In particular, the interaction between the vortices is ruled by the Green function of the Laplacian in \(\mathbb {R}^2\) .
On the contrary, when dealing with the energy \(\frac{1}{2}\int _{M}\left| D u \right|^2_g \mathrm {vol}_g\) , we have to deal with the Laplace-Beltrami operator on \(M\)  and with its Green function.
The resulting renormalized energy inherits the complexity of the Green function (see [9]})
and couples the Gauss curvature of \(M\)  with the position and with the charge of the vortices. Moreover, it explicitly depends on the harmonic 1-form \(\xi \) .
Finally,
the interaction term between the positions of the vortices behaves logarithmically only when the geodesic distance between the two vortices is small enough.
